$\displaystyle \frac{\partial ^2}{\partial x^2}\left(\frac{xy}{2x+y}\right)$
using the quotient rule and treating $y$ as a constant we get
$\dfrac{\partial}{\partial x}\left(\dfrac{x y}{2x+y}\right) = \dfrac{y(2x+y) - 2xy}{(2x+y)^2} = \left(\dfrac{y}{2x+y}\right)^2$
$\dfrac{\partial}{\partial x}\left(\dfrac{y}{2x+y}\right)^2 = 2\left(\dfrac{y}{2x+y}\right)\cdot \left(\dfrac{-y}{(2x+y)^2}\right)\cdot 2 = -\dfrac{4y^2}{(2x+y)^3}$
$\dfrac{\partial^2}{\partial x^2}\left(\dfrac{x y}{2x+y}\right) = -\dfrac{4y^2}{(2x+y)^3}$